\subsection{Delay Effects}

\subsubsection{Echo}
Delays and echoes are essentially the same, when talking about audio. A delay implies that some sort of audio input recorded and then played back after some period of time. An echo does the same, but often this is done repeatedly with attenuated volume and/or quality for each step to simulate the effect of a decaying echo in e.g. inside a cave.

In other words, an echo is a copy of the original audio signal that is delayed by a fixed amount in time.\cite{delay_website}.

\textit{Reverb}, \textit{vibrato}, \textit{flanger}, \textit{chorus} and \textit{echo} are examples of delay effects. The difference between echo and reverb is that an echo has a distinct, delayed version of the audio, while reverb delivers the delayed audio in short periods of time that makes it difficult to distinguish each delay.

The delay/echo is done by having a memory buffer that records a series of samples. These are then played back after a set amount of time. Sometimes the original signal and the modified (delayed) signal are mixed together.

The key parameter here is the delay time; how long should the input be recorded before playing it. Other parameters are amplification of the repeated signal and whether it should use feedback of feed forward.

A simple delay filter can be expressed as $y_t = x_t + a_1 *x_{t-\tau}$. Here we start with a signal, $x$, and add to it some constant, $a_1$, times a delayed version of itself, $x_{t-\tau}$.

Figure \ref{fig:delay} shows this as well. The input enters from the left. The delayed version is then obtained by tapping it at the junction that is indicated by the small dot and goes to the box labelled with the delayed $\tau$. The addition of the signal to the delayed version of itself is shown in the conjugation with the summation symbol $\sum$.\cite{DSPPrimer}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.3\textwidth]{images/TheoryDesign/delay_simple}
\caption{A simple feedforward delay.\cite{DSPPrimer}}
\label{fig:delay}
\end{figure}